On parabolic Adams's, the Chiarenza-Frasca theorems, and some other results related to parabolic Morrey spaces

1.   Introduction

Let $\mathbb{R}^{d}$, $d\geq 1$, be a Euclidean space of points $x = (x^{1}, ..., x^{d})$. In 1975 D. Adams ^[1] among many other things proved that, if $d\geq2$ and we are given $u\in C^{\infty}_{0} = C^{\infty}_{0}(\mathbb{R}^{d})$ with its gradient $Du = (D_{1}u, ..., D_{d}u)$, $D_{i} = \partial/\partial x^{i}$, satisfying

with $q > 1$, $1 < \beta\leq d/q$, and any $\rho\in(0, \infty)$ and $x\in \mathbb{R}^{d}$, then for all $\rho\in(0, \infty)$ and $x\in \mathbb{R}^{d}$ we have

with a constant $N$ independent of $u$ and $r$ satisfying $(\beta-1)r = \beta q$.

This fact played a crucial role in ^[10] where the author investigated the solvability of elliptic equations

with $b\not \in L_{d, \text{loc}}$ but rather satisfying for a sufficiently small $\hat b$, all sufficiently small $\rho$ and all balls $B$ of radius $\rho$

with certain $d_{0}\in(d/2, d)$.

Our goal in this paper is to prepare necessary tools for developing a similar theory for parabolic equations. In Section 2 we prove an analog of Adams's intermediate estimate, which is the main starting point. Section 3 contains the parabolic analog of the embedding theorem mentioned in the beginning of the article. It also contains "local" interpolation inequalities in Morrey spaces allowing one to deal with Morrey's norms of expressions like $b^{i}D_{i}u$ in domains when $b$ is bounded. Section 4 is devoted to the parabolic analog of a Chiarenza-Frasca theorem allowing to estimate the $L_{p}$-norm rather than Morrey's norm of $b^{i}D_{i}u$. In Section 5 we treat parabolic Morrey spaces with mixed norms. The main object of investigation is the term $b^{i}D_{i}u$ and the ways to estimate it in various Morrey and $L_{p}$ spaces in order to be able to treat it as a perturbation term in the parabolic analog of (1.3).

We finish the introduction with some notation and a remark. Define $B_{\rho}(x) = \{y \in \mathbb{R}^{d} :|x-y| < \rho\}$, $\mathbb{R}^{d+1}: = \{z = (t, x):t\in \mathbb{R}, x\in \mathbb{R}^{d}\}$,

and let $\mathcal{C}_{\rho}$ be the collection of $C_{\rho}(z)$, $z\in \mathbb{R}^{d+1}$, $\mathcal{C} = \{ \mathcal{C}_{\rho}, \rho > 0\}$. For measurable $\Gamma\subset \mathbb{R}^{d+1}$ set $|\Gamma|$ to be its Lebesgue measure and when it makes sense set

Similar notation is used for $f = f(x)$.

Remark 1.1. Formally, Adams proved (1.2) assuming that $d\geq2$. However, it is also true if $d = 1$. To show this it suffices to take $u$ depending only on one coordinate. The reader may wonder how the restriction $\beta\leq d/q$ will become $\beta\leq 1/q$. The point is that if $d = 1$ and $\beta > 1/q$, we have $d-\beta q < 0$ and condition (1.1) becomes only possible if $u = 0$.

2.   Preliminary estimates

An important quantity characterizing $L_{p} = L_{p}(\mathbb{R}^{d+1})$ is what we call the index which is the exponent of $\rho$ in the expression

For domains $Q\subset \mathbb{R}^{d +1 }$, $p\in[1, \infty)$, and $\beta\in(0, (d+2)/p]$, introduce Morrey's space $E_{p, \beta}(Q)$ as the set of $g$ such that

where

We abbreviate $E_{p, \beta} = E_{p, \beta}(\mathbb{R}^{d+1})$. Observe that if $Q = C_{R}$ one can restrict $\rho$ in (2.1) to $\rho\leq R$ since $\beta\leq (d+2)/p$. Also in that case one can allow $(t, x)$ to be arbitrary, because, if $|x|\geq R$, then $B_{R}\cap B_{\rho}(x)\subset B_{R}\cap B_{\rho}(Rx/|x|)$. It is also useful to observe that, in case $Q = C_{R}$, one gets an equivalent norm by adding to the restrictions $\rho < \infty, (t, x)\in C_{R}$, the requirement that the geometric center of $C_{\rho}(t, x)$ be in $C_{R}$. This follows from the fact that the $L_{p}(C_{\rho}(t, x))$-norm of $gI_{C_{R}}$ will only increase if we pull $C_{\rho}(t, x)$ down the $t$ axis to $\{t = 0\}$ (if $\rho^{2} > 2R^{2}$) or to the moment that the shifted $C_{\rho}(t, x)$ has its geometric center inside $C_{R}$.

There are many different notations for the norms in Morrey spaces. The convenience of the above notation is well illustrated by Theorem 3.1 and Corollary 5.7.

We will often, always tacitly, use the following formulas in which $u(t, x) = v(t/R^{2}, x/R)$:

where $Q_{R} = \{(t, x): (R^{2}t, Rx)\in Q\}$,

For $s, r > 0, \alpha > 0$, and appropriate $f(t, x)$'s on $\mathbb{R}^{d+1}$ define

Observe that, if $f$ is independent of $t$, then

where $I_{\alpha}$ is the Riesz potential. Therefore, one can get the Adams estimate found in the proof of Proposition 3.1 of ^[1] from (2.3) below. In our investigation the most important values of $\alpha$ are 1 and 2. Set

The following lemma is obtained by integrating by parts.

Lemma 2.1. Let $\beta > 0$ be a finite number, $f(t)\geq0$ be a function on $[0, \infty)$ such that

as $t\to \infty$. Then, for any $S\geq 0$,

Lemma 2.2. For any $\alpha\in(0, \beta), \beta\in(0, d+2]$ there exist constants $N$($< \infty$) such that for any $f\geq0$ and $\rho\in(0, \infty)$ we have

In particular (by Hölder's inequality), for any $p\in[1, \infty]$, $q\in(1, \infty]$, and measurable $\Gamma$

provided that

Proof. We basically mimic the proof of Proposition 3.1 of ^[1]. Observe that (2.3) at the origin is easily obtained from summing up the inequalities in (2.2) and minimizing with respect to $\rho$. At any other point it is obtained by changing the origin. Furthermore clearly, we may assume that $f$ is bounded with compact support. Set $Q_{1} = \{(s, y):|y|\geq\sqrt s\}$, $Q_{2} = \{(s, y):|y|\leq \sqrt s\}$. Dealing with $P_{\alpha}(fI_{Q_{1}})$ we observe that $p_{\alpha}(s, r)\leq Nr^{-(d+2-\alpha)}$ if $r\geq\sqrt s$. Therefore,

where $d\sigma_{r}$ is the element of the surface area on $|y| = r$. By Lemma 2.1 ($\alpha < d+2$)

where

We use that $I(r)\leq Nr^{ d+2-\beta }M_{\beta}f(0)$ and that $\alpha < \beta$. Then we see that

Next, by using Lemma 2.1 we obtain that

This along with (2.5) prove the second inequality in (2.2).

As long as the first inequality is concerned, observe that similarly to Lemma 2.1 using that $\alpha > 0$ we have

where

Here $I(r) \leq Nr^{ d+2 } Mf(0)$ and $\alpha > 0$, so that

Furthermore,

where

This and (2.6) prove the first inequality in (2.2). The lemma is proved.

Remark 2.3. If $d = \alpha = 1$ and $f$ is independent of $t$, the inequalities (2.2) and (2.3) are useless, because the first one in (2.2) follows by definition and the second one and (2.3) are trivial because $M_{\beta}f = \infty$ ($\beta > \alpha = 1$) unless $f = 0$.

If $\alpha$ is strictly less than the index of $L_{q}$, we have the following.

Corollary 2.4. If $\alpha\in(0, (d+2)/q)$, $q\in(1, \infty)$, thenthere exists a constant $N$ such that for any $f\geq0$ we have

as long as

In particular, (a classical embedding result) if $1 < q < d+2$ and $u\in C^{\infty}_{0} = C^{\infty}_{0}(\mathbb{R}^{d+1})$, then

as long as

Indeed, the first assertion follows from Hölder's inequality and (2.4) with $p = \infty$ and $\beta = (d+2)/q$ ($> \alpha$). The second assertion follows from the first one with $\alpha = 1$ ($< \beta$) and the fact that for $f = \partial_{t}u+\Delta u$ we have

where $c$ is a constant and $(|y|/s^{1/2})e^{-|y|^{2}/(4s)}\leq Ne^{-|y|^{2}/(8s)}$.

Remark 2.5. After Corollary 2.4 a natural question arises as to what power of summability $b = (b^{i})$ will be sufficient for the term $b^{i}D_{i}u$ to be considered as a perturbation term in $\partial_{t}u+\Delta u +b^{i}D_{i}u$ in the framework of the $L_{q}$-theory. Observe that, in the notation of Corollary 2.4

It follows that $b$ should be of class $L_{d+2}$ and $q < d+2$. Of course, if $b$ contains just bounded part, this part in $b^{i}D_{i}u$ is taken care of by interpolation inequalities.

In the next section we will also need the following result.

Corollary 2.6. For any $\alpha\in(0, \beta), \beta\in(0, d+2]$ there exists a constant $N$ such that for any $g\geq0$, $\rho\in(0, \infty)$, and $(t, x)\in C_{\rho}$ we have

Indeed, since

for $f = g(t+\cdot, x+\cdot)$ we have

3.   A parabolic analog of the Adams Theorem 3.1 of ^[1]

Theorem 3.1. For any $\alpha\in(0, \beta), \beta\in(0, (d+2)/q]$, $q\in(1, \infty)$, and $r$ such that

there is a constant $N$ such that for any $f\geq0$ we have

Proof. It suffices to prove that for any $\rho > 0$

that is

Observe that by Hölder's inequality $M_{\beta}f\leq N\|f\|_{E_{q, \beta}}$ and by definition

It follows from Lemma 2.2 with $p = \infty$ that

Furthermore, by Corollary 2.6

By combining these estimates we come to (3.2) and the theorem is proved.

Remark 3.2. We did not explicitly used that $\beta \leq (d+2)/q$ and formally the proof is valid for any $\beta\in(0, \infty)$ if in Definition 2.1 we allow any $\beta > 0$. However, if $\beta > (d+2)/q$ and $f\ne 0$, the right-hand side of (3.1) is infinite. Therefore, to make Theorem 3.1 nontrivial one requires $\beta\leq (d+2)/q$.

Remark 3.3. There is a simple relation of $P_{\alpha_{1}} P_{\alpha_{2}}$ to $P_{\alpha_{1}+\alpha_{2}}$, which, in light of Theorem 3.1, implies that, if $\beta > \alpha_{2}\geq\alpha_{1} > 0$, $q_{1}, q_{2}\in(1, \infty)$, $q_{1}(\beta-\alpha_{1}) = q_{2}(\beta-\alpha_{2}) \leq d+2$, then $\|P_{\alpha_{2}}f\|_{E_{q_{2}, \beta-\alpha_{2}}} \leq N\|P_{\alpha_{1}}f\|_{E_{q_{1}, \beta-\alpha_{1}}}$. We leave details of the proof to the reader and we do not use this fact in what follows.

The following, obtained similarly to Corollary 2.4, was communicated to the author by Hongjie Dong.

Corollary 3.4. If $1 < q < d+2$, $\beta\in(1, (d+2)/q]$, and $u\in C^{\infty}_{0}$, then

as long as

Remark 3.5. For $\beta = (d+2)/q$ Corollary 3.4 yields the second part of Corollary 2.4 once more. This is because $E_{q, (d+2)/q} = L_{q}$.

Remark 3.6. In the framework of the Morrey spaces Corollary 3.4 opens up the possibility to treat the terms like $b^{i}D_{i}u$ as perturbation terms in operators like $\partial_{t}u+\Delta u +b^{i}D_{i}u$ even with rather low summability properties of $b = (b^{i})$. To show this, observe that for $f, g\geq0$ in the notation of Corollary 3.4

It follows that

For $\beta = (d+2)/q$ estimate (3.4) coincide with (2.7), but for $\beta < (d+2)/q$ in the framework of Morrey spaces we allow $b$ to be summable to the power $\beta q < d+2$ in contrast with Remark 2.5. However, we need $\|b \|_{E_{\beta q, 1}} < \infty$ and, if we ask ourselves what $r$ should be in order for $b\in L_{r}$ to have $\|b \|_{E_{\beta q, 1}} < \infty$, the answer is $r = d+2$ at least. Still we gain the possibility to have higher singularities of $b$ than functions from $L_{d+2}$. Elliptic versions of (3.4) for usual or generalized Morrey spaces are found in many papers, see, for instance, ^[5] and the references therein.

Next we move to deriving "local" versions of the above results. A statement somewhat weaker than Corollary 3.4 can be obtained from the following general result by taking $(S, T)$ to be large enough and then sending $S\to-\infty, T\to\infty$.

Theorem 3.7. Let $1 < q < d+2$, $\beta\in(1, (d+2)/q]$ and let $(3.3)$ hold. Thenthere is a constant $N$ such thatfor any $u\in C^{\infty}_{0}$, $-\infty < S < T < \infty$, and $Q_{S, T} = (S, T)\times \mathbb{R}^{d}$

Proof. Shifting and changing the scales in $\mathbb{R}^{d+1}$ allow us to assume that $S = -1 = -T$. In that case consider the mapping $\Phi:[-3/2, 3/2]\to[-1, 1]$, $\Phi(t) = t\big(2/(|t|\vee 1)-1\big)$ that preserves $[-1, 1]$, is Lipschitz continuous and has Lipschitz continuous inverse if restricted to $[-3/2, 3/2]\setminus (-1, 1)$. Then, obviously, for $w(t, x) = v(\Phi (t), x)$ we have

where $N = N(q)$.

Now take $(t, x)\in Q_{-1, 1}$, $\rho\in(0, \infty)$, and take $\zeta\in C^{\infty}_{0}(\mathbb{R})$ such that $\zeta = 1$ on $(-1, 1)$, $\zeta = 0$ outside $(-3/2, 3/2)$, and $|\zeta|+| \zeta'| \leq 4$.

Although the function $\zeta w$, where $w(s, y) = u(\Phi(s), y)$, is not as smooth as required in Corollary 3.4 the argument leading to it applies to $\zeta(s)w(s, y)$ (we have a general Remark 5.14 to that effect) and since $r(\beta-1) = q\beta$ we have

It only remains to note that the last expression is less than the right-hand side of (3.5) in light of (3.6). The theorem is proved.

To prove an interpolation theorem in $C_{R}$ we need two lemmas.

Lemma 3.8. Let $0 < R_{1} < 1 < R_{2} < \infty$, $1\leq q < \infty$, $\beta\in(0, (d+2)/q]$. Define $\Gamma_{1} = \bar B_{1}\setminus B_{R_{1}}$, $\Gamma_{2} = \bar B_{R_{2}}\setminus B_{1}$ and let $\Phi:\Gamma_{2}\to \Gamma_{1}$ be a smoothone-to-one mapping with $|D\Phi|, |D\Phi^{-1}|\leq K$, where $K$ is a constant. Let $v(t, x)\geq0$ be zero outside $G_{2}: = (0, 1)\times\Gamma_{2}$ and set $u(t, x) = v(t, \Phi^{-1}(x))I_{\Gamma_{1}}(x)$. Then

Proof. Take $(t, x)\in (0, 1)\times B_{R_{2}}$ and $\rho > 0$. Then

where $\Psi(s, y) = (s, \Phi(y))$. Observe that, if $C_{\rho}(t, x)\cap G_{2} \ne\emptyset$, then $|y_{1}-y_{2}|\leq 2\rho$ for any $y_{1}, y_{2}\in C_{\rho}(t, x)\cap G_{2}$. It follows that $\Phi(C_{\rho}(t, x)\cap G_{2})\subset B$, where $B$ is a ball of radius $2K\rho$ with center in $B_{1}$, and

This proves the lemma.

The following lemma about the interpolation inequality (3.9) is quite natural and obviously useful, but its elliptic counterpart was proved only rather late in ^[10]. One of its goals is to be able to treat $b^{i}D_{i}u$, when $b$ is bounded, as a perturbation term.

Lemma 3.9. Let $p\in(1, \infty)$, $0 < \beta\leq (d+2)/p$. Then there is a constant $N$ such that, for any $R\in(0, \infty)$, $\rho\leq 2R$, $C\in \mathcal{C}_{\rho}$ with its geometric centerin $C_{R}$, $\varepsilon\in(0, 1]$, and $u\in C^{\infty}_{0}$, we have

where $c$ is any constant and $C(s)\in \mathcal{C}_{s}$ with the geometric centerthe same as $C$. In particular,

Proof. Changing scales shows that we may assume that $R = 1$. Obviously we may also assume that $c = 0$. Then denote $v = Du$, $w = |\partial_{t}u|+|D^{2}u|$, $G_{s} = C(s) \cap C_{1}$,

By Poincaré's inequality (see, for instance, Lemma 5.9), for $\rho\leq s\leq 2$,

Also by interpolation inequalities, there exists a constant $N = N(d, p)$ such that, for $\varepsilon \in(0, 1]$ and $\varepsilon\leq s\leq 2$,

which for $2\geq s\geq \varepsilon\vee \rho$ yields

Hence, for any $\varepsilon\in(0, 1]$ and $\rho\leq s\leq 2$

where $N_{1} = N_{1}(d, p)$, $N_{2} = N_{2}(d, p)$.

Following Campanato, one can transform this result to estimate $v_{G_{s}}$ going along $\rho$, $2\rho$, ... and, since $\beta\in(0, (d+2)/p]$, by Campanato's results (cf. for instance, Proposition 5.4 in ^[8]) one gets that

where $N_{3} = N_{3}(d, p, \beta)$. We estimate the last term as in (3.10) and come to what implies (3.8). The lemma is proved.

The following is a local version of Corollary 3.4. It allows us to draw the same conclusions as in Remark 3.6 in bounded domains.

Theorem 3.10. Let $1 < q < d+2$, $\beta\in(1, (d+2)/q]$ and let $r(\beta-1) = q\beta$. Then there is a constant $N$ such thatfor any $R\in(0, \infty]$, $u\in C^{\infty}_{0}$,

Proof. The case of $R = \infty$ is obtained by passing to the limit. In case $R < \infty$, as usual, we may assume that $R = 1$. In that case, mimicking the Hestenes formula, for $1\leq |x|\leq 6/5$ define

and for $|x|\leq 1$ set $v(t, x) = u(t, x)$. One can easily check that $v\in C^{1, 2}([0, 1] \times B_{6/5})$. In light of Lemmas 3.8 and 3.9, for instance,

Now take $(t, x) \in C_{1}$, $\rho\in(0, \infty)$, and take $\zeta\in C^{\infty}_{0}(\mathbb{R}^{d})$ such that $\zeta = 1$ on $B_{1}$, $\zeta = 0$ outside $B_{6/5}$, and $|\zeta|+|D\zeta| +|D^{2}\zeta|\leq N = N(d)$.

By using Theorem 3.7 we get

It only remains to note that the last expression is less than the right-hand side of (3.11) as is well seen from (3.12). The theorem is proved.

Remark 3.11. By considering functions depending only on $x$ we naturally obtain "elliptic" analogs of our results. For instance, for $G\subset \mathbb{R}^{d}$ by defining

we get from (3.11) for $u\in C^{\infty}_{0}(\mathbb{R}^{d})$ that

whenever $1 < q < d, \beta\in(1, d/q]$ and $r(\beta-1) = q\beta$. Actually, formally, one gets (3.13) even for $\beta\leq (d+2)/q$, but for $\beta > d/q$, both sides of (3.13) are infinite unless $u = 0$.

After that arguing as in (3.4) we see that for $1 < q < d, \beta\in(1, d/q]$

From the point of view of the theory of elliptic equations the most desirable version of (3.14) would be

for any $\varepsilon > 0$ with $N(\varepsilon)$ independent of $u$. This fact is, actually, claimed in Theorem 5.4 of ^[5]. We will show that (3.15) cannot hold if $\varepsilon$ is small enough.

Let $h(t)$ be a smooth nondecreasing function on $\mathbb{R}$ such that $h(t) = 0$ for $t\leq 0$, $h(t) = t$ for $t\geq 1$ and for $\delta > 0$ set $u_{\delta}(x) = h(\ln(\delta/|x|))$. Let $1 < q < d/2$, $\beta = 2$, $b(x) = 1/|x|$.

Then

as $\delta\downarrow0$. At the same time

It is seen that $|D^{2}u_{\delta}|\leq N(d)/|x|^{2}$ and, since $q < d/2$, the $E_{q, \beta}(B_{1})$-norm of $D^{2}u_{\delta}$ is bounded as $\delta \downarrow 0$. Also, for $|x|\leq\delta/e$, we have $b|Du_{\delta}| = 1/|x|^{2}$, so that for $r\leq\delta/e$

It follows that the $E_{q, \beta}(B_{1})$-norm of $b|Du_{\delta}|$ is bounded away from zero as $\delta \downarrow 0$ and this shows that (3.15) cannot hold for all $\delta > 0$ if $\varepsilon$ is small enough.

4.   A parabolic version of Chiarenza–Frasca result ^[4]

In Remark 3.6 we have shown how to estimate a Morrey norm of $|b|\, |Du|$ in terms of a Morrey norm of $b$. Here, following ^[4], we show how to estimate an $L_{p}$-norm of the same quantity through the $L_{p}$-norms of $\partial_{t}u$ and $D^{2} u$.

Theorem 4.1. Let $d+2\geq q > p > 1$, $b\in E_{q, 1}$. Then for any $f\geq0$ we have

where $N$ depends only on $d, p, q$.In particular (see the proof ofCorollary 2.4), for any $u\in C^{\infty}_{0}$

where $K = \|D^{2}u, \partial_{t}u\|^{p}_{L_{p}}$ and $N$ depends only on $d, p, q$.

Observe that we already know this result if $q = d+2$ from Remarks 2.5 or 3.6.

In the proof we are going to use "parabolic" versions of some results from Real Analysis associated with balls and cubes. These versions are obtained by easy adaptation of the corresponding arguments by replacing balls with parabolic cylinders and cubes with parabolic boxes. To make the adaptation more natural we introduce the "symmetric" maximal parabolic function operator by

where (recall that) $\mathcal{C}$ is the set of $C_{r}(z)$, $r > 0$, $z\in \mathbb{R}^{d+1}$. To prove the theorem we need the following.

Lemma 4.2. a) For $r\in(0, \infty)$ define $D _{r} = \{|t|\leq r^{2}, |x|\leq r \}$. Then

where $N = N(d)$.

b) For any nonnegative $g(t, x)$, $q\in[1, \infty)$, $\beta\in(0, d+2]$, $\alpha > 0$, $\alpha > 1-q\beta/(d+2)$, and $r\in(0, \infty)$

Proof. Assertion a) is proved by elementary means. To prove b), we use a) and split $D^{c}_{2r}$ into two parts $D^{c}_{2r}\cap\{|x|^{2}\geq |t|\}$ and $D^{c}_{2r}\cap\{|x|^{2} < |t|\}$ and, taking into account obvious symmetries, we see that it suffices to show that

By observing that

as $t\to\infty$, we have

Also as is easy to see

This proves the lemma.

Proof of Theorem 4.1. We follow some arguments in ^[4] and may assume that $b\geq0$. First set $r_{0} = (p+q)/2$ and assume that there is a constant $N_{0}$ such that $\hat M(|b|^{r_{0}})\leq N_{0}|b|^{r_{0}}$, that is, $|b|^{r_{0}}$ is in the class $A_{1}$ of Muckenhoupt. Observe that by Hölder's inequality $\|b\|_{E_{r_{0}, 1}}\leq \|b\|_{E_{q, 1}}$. It is convenient to prove the following version of (4.1) (notice $r_{0}$ in place of $q$)

Then assume that $b\geq 0$, set $u = P_{1}f$, and write

where $p' = p/(p-1)$ and $P^{*}_{1}$ is the conjugate operator for $P_{1}$, namely, for any $g\geq0$,

Next, take $\gamma > 0$, such that $(1+\gamma)p \leq r_{0}$, $1+\gamma p'\leq r_{0}$, and $p\geq 1+\gamma$. Note that

It follows that

Now in light of (4.6) we see that, to prove (4.5) in our particular case, it only remains to show that

For $\alpha = 1$ and $\beta = (1+\gamma)p$ ($> \alpha$) it follows from (2.3) and (4.7) that

where the last factor by assumption (and Hölder's inequality) is dominated by $Nb^{(1+\gamma)p-1}$ and $\|b\|_{E_{\beta, 1}}\leq \|b\|_{E_{r_{0}, 1}}$. After that to obtain (4.8) it suffices to use again (2.3) with $\alpha = 1$ and $\beta = 1+\gamma p'$ to see that

We now get rid of the assumption that $\hat M(|b|^{ r_{0} })\leq N_{0}|b|^{ r_{0 }}$ as in ^[4].

For $r_{1} = (r_{0} +q)/2$ we have $|b|^{ r_{0} }\leq (\hat M(|b|^{ r_{1} }))^{ r_{0 }/ r_{1}} : = \tilde b^{ r_{0} }$ and since $r_{0} / r_{1} < 1$, $\tilde b^{ r_{0} }$ is an $A_{1}$-weight with $N_{0} = N_{0}(r_{0} / r_{1})$ (see, for instance, ^[7] p. 158). Therefore, (4.5) holds with $\tilde b$ in place of $b$ and it only remains to show that

that is, for any $t, x, \rho$,

Of course, we may assume that $t = 0, x = 0$. Then by Hölder's inequality we see that the left-hand side of (4.9) is less than

where the integral by a Fefferman-Stein Lemma 1, p. 111 of ^[6] and the fact that $q/ r_{1 } > 1$ is dominated by

where we used Lemma 4.2 b) for $\alpha = \beta = 1$. Hence,

which is (4.9).

An alternative way to get the result is to follow the proof of Theorem 3 of ^[3]. We have

where $\alpha\in(0, 1)$. An easy exercise leads to the well-known result that $(\hat MI_{C_{\rho}})^{\alpha}$ is an $A_{1}$-weight, and, hence, an $A_{q/ r_{0} }$-weight. By the Muckenhoupt theorem

and it only remains to use Lemma 4.2 b) again with $\beta = 1$ and any appropriate $\alpha$. The theorem is proved.

Remark 4.3. In the above proof we tacitly assumed that $I < \infty$. One can easily avoid it by taking $f$ with compact support, replacing $|b|$ with $|b|\wedge b_{n}$, where $n^{-1}b_{n} = 1\wedge (\sqrt{|t|}+|x|)^{-1}$, observe that $b^{r_{0}}_{n}\in A_{1}$, and while checking that the new $I$ is finite use Hölder's inequality and Corollary 2.4.

5.   Mixed-norm estimates

For $q_{1}, q_{2}\in[1, \infty]$ and measurable $f$ and $\Gamma\subset \mathbb{R}^{d+1}$ introduce

Here the index of $L_{q_{1}, q_{2}}$ which is the exponent of $\rho$ in the expression

If in addition $0 < \beta \leq d/q_{1}+2/q_{2}$, set

We also introduce the spaces $L_{q_{1}, q_{2}}(Q)$ and $E_{q_{1}, q_{2}, \beta}(Q)$ as the spaces of functions whose respective norms are finite. We abbreviate $L_{q_{1}, q_{2}} = L_{q_{1}, q_{2}} (\mathbb{R}^{d+1})$, $E_{q_{1}, q_{2}, \beta} = E_{q_{1}, q_{2}, \beta} (\mathbb{R}^{d+1})$.

The following is certainly well known.

Lemma 5.1. Let $f$ be a nonnegative function on $\mathbb{R}^{d+1}$, $p, q\in(1, \infty)$. Then forany $w_{x}(x), w_{t}(t)$ which are $A_{p}$ Muckenhoupt weights on $\mathbb{R}^{d}$ and $\mathbb{R}$, respectively, we have

where $N$ depends only on $d, p$, and the $A_{p}$-constants of $w_{x}, w_{t}$. Furthermore,

where $N$ depends only on $d, p, q$.

Proof. Estimate (5.1) follows by application of the Muckenhoupt theorem to $w_{x}w_{t}$, which is an $A_{p}$-weight on $\mathbb{R}^{d+1}$. Then observe that in the particular case that $w_{x}\equiv 1$, (5.1) means that

for any $A_{p}$-weight $w_{t}$, which implies (5.2) by the Rubio de Francia extrapolation theorem. The lemma is proved.

This lemma, (2.3), and Hölder's inequality immediately yield the following.

Lemma 5.2. For any $\alpha\in(0, \beta), \beta\in(0, d+2]$, $p\in[1, \infty]$, $q_{1}, q_{2}\in(1, \infty]$, there exists a constant $N$ such that for any $f\geq0$ and measurable $\Gamma$ we have

provided that

Similarly to Corollary 2.4 we have

Corollary 5.3. Let $q_{1}, q_{2}\in(1, \infty]$,

$\alpha\in(0, \beta)$. Then for any $f\geq0$ we have

as long as $q_{i}\beta = r_{i}(\beta-\alpha)$, $i = 1, 2$.

In particular, (almost follows fromTheorem 10.2 of ^[2])if $\beta > 1$, and $u\in C^{\infty}_{0}$, then

as long as $q_{i}\beta = r_{i}(\beta-1)$, $i = 1, 2$.

Corollary 5.4. Under the assumptions of Corollary5.3, if $\beta > 1$, there is a constant $N$ such that, for any $b = (b^{i})\in L_{\beta q_{1}, \beta q_{2}}$ and $u\in C^{\infty}_{0}$,

Indeed, by Hölder's inequality

Remark 5.5. It is instructive to compare this result with Remark 2.5. Now we can treat $b\in L_{s_{1}, s_{2}}$ with $s_{i}\in(1, \infty]$ satisfying $d/s_{1} +2/s_{2} = 1$.

Since $E_{q_{1}, q_{2}, \beta} = L_{q_{1}, q_{2}}$ if $\beta = d/q_{1}+2/q_{2}$, the following is a generalization of Corollary 5.3.

Theorem 5.6. Let $q_{1}, q_{2}\in(1, \infty]$,

$\alpha\in(0, \beta)$. Then there is a constant $N$ such that for any $f\geq0$ we have

where $r_{i}(\beta-\alpha) = q_{i}\beta$, $i = 1, 2$.

Proof. It suffices to prove that for any $\rho > 0$

that is

Observe that by Hölder's inequality $M_{\beta}f\leq N\|f\|_{E_{q_{1}, q_{2}, \beta}}$ and by definition

It follows from Lemma 5.2 with $p = \infty$ that (5.7) holds with $I_{C_{2\rho}}f$ in place of $f$ on the left.

Furthermore, by Corollary 2.6 we have $|P_{\alpha}(I_{C_{2\rho}^{c}}f)| \leq N\rho^{\alpha-\beta}M_{\beta}f$ in $C_{\rho}$. Therefore,

By combining these results we come to (3.2) and the theorem is proved.

Corollary 5.7. Under the assumptions of Theorem 5.6, if $\beta > 1$, for any $u\in C^{\infty}_{0}$, we have

where $r_{i}(\beta-1) = q_{i}\beta$, $i = 1, 2$. This coincides with $(5.4)$ if $\beta$ is equal to the index of $L_{q_{1}, q_{2}}$.

Remark 5.8. Corollary 5.7 opens up the possibility to treat the terms like $b^{i}D_{i}u$ as perturbation terms in operators like $\partial_{t}u+\Delta u +b^{i}D_{i}u$ with even lower summability properties of $b = (b^{i})$ than in Remark 5.5. To show this observe that for $q_{1}, q_{2}, \beta$ as in Theorem 5.6 with $\beta > 1$ and $s_{i} = \beta q_{i}\in(1, \infty]$, $i = 1, 2$, we have

implying that

where $d/s_{1}+2/s_{2}\geq 1$.

However, note that we also need

to be bounded as a function of $\rho, t, x$. If we ask ourselves what $\tau > 0$ should be to guarantee this boundedness if $b\in L_{\tau s_{1}, \tau s_{2}}$, if $d/s_{1}+2/s_{2} > 1$, the slightly disappointing answer is that $\tau = d/s_{1}+2/s_{2}$, so that $d/(\tau s_{1})+2/(\tau s_{2}) = 1$. Still functions in $E_{s_{1}, s_{2}, 1}$ may have higher singularities than those in $L_{\tau s_{1}, \tau s_{2}}$.

Another advantage of (5.8) in comparison with (5.5) is seen when $b$ depends only on $t$ or $|b(t, x)|\leq \hat b(t)$. In that case (5.8) becomes

and if $\beta q_{2} = 2$, then

Thus for any $q_{1}\in(1, \infty]$ and $q_{2}\in(1, 2)$

In case $q_{1}\in(1, d)$, $q_{2}\in(1, \infty]$, $1 < \beta\leq d/q_{1}$, and $\|b(\cdot, t)\|_{E_{\beta q_{1}, 1}(\mathbb{R}^{d})}\leq \hat b < \infty$ for any $t$, we also have

An application of the last inequality in case $u, b$ are independent of $t$, $\beta = d/q_{1}$, $q_{1}\in(1, d)$, and $q_{2} = \infty$, yields the well-known estimate

To extend the embedding and interpolation results to Morrey spaces with mixed norms we need the following result very useful also in other circumstances.

Lemma 5.9 (Poincaré's inequality). Let $1\leq r_{1}, r_{2} < \infty$, $u\in C^{\infty}_{0}$, $\rho\in(0, \infty)$.Then

Proof. We follow the usual way (see, for instance, Lemma 4.2.2 of ^[9]). First, due to self-similar transformations, we may take $\rho = 1$. In that case, for a $\zeta\in C^{\infty}_{0}(B_{1})$ with unit integral, introduce

Then by the usual Poincaré inequality

Next, observe that for any constant vector $v$ the left-hand side of (5.9) is less than a constant times (recall that $\rho = 1$)

By (5.10) the first term on the right is less than the right-hand side of (5.9). To estimate the second term, take

Then by Poincaré's inequality

and to finish the proof it only remains to use Hölder's inequality. The lemma is proved.

The usual Poincaré inequality was used in the proof of Lemma 3.9. Also observe that mixed-norms estimates like (3.10) are available in ^[2] (see Theorem 9.5 there). Therefore, by using Lemma 5.9 and following very closely the proofs of Lemmas 3.8, 3.9, and Theorems 3.10 we arrive at the following results about interpolation and embedding for Morrey spaces with mixed norms.

Lemma 5.10. Let $q_{1}, q_{2}\in(1, \infty)$, $0 < \beta\leq d/q_{1}+2/q_{2}$. Then there is a constant $N$ such that, for any $R\in(0, \infty)$, $\varepsilon\in(0, 1]$, and $u\in C^{\infty}_{0}$,

Theorem 5.11. Let $q_{1}, q_{2}\in(1, \infty)$, $1 < \beta\leq d/q_{1}+2/q_{2}$ and let $r_{i}(\beta-1) = q_{i}\beta$, $i = 1, 2$. Then there is a constant $N$ such that for any $R\in(0, \infty]$, $u\in C^{\infty}_{0}$ we have

Remark 5.12. By taking $u$ depending only on $x$ we recover from Lemma 5.10 and Theorem 5.11 their "elliptic" counterpart stated as Lemmas 4.4 and 4.7 in ^[10], respectively.

Remark 5.13. Theorem 5.11 is the most general results of the paper containing as particular cases our previous results on embeddings. Thus, Corollary 5.7 (in an obvious rougher form) follows from Theorem 5.11 when $R = \infty$ and contains embedding results for Lebesgue spaces with mixed norms as $\beta = d/q_{1}+2/q_{2}$ and for $L_{q}$-spaces as $q = q_{1} = q_{2}$.

Remark 5.14. We stated our results only for $u\in C^{\infty}_{0}$ just for convenience. Let us show why, for instance, Theorem 5.11 is valid as long as $\partial_{t}u, Du, D^{2}u\in E_{q_{1}, q_{2}, \beta}(C_{R})$. For that, it suffices to prove that for any $R' < R$, $\rho > 0$, $(t, x) \in C_{R'}$ the quantity

is less than the right-hand side of (5.12) with $(R')^{-2}$ in place of $R^{-2}$. For $\varepsilon > 0$ define $u^{(\varepsilon)} = (I_{C_{R}}u)*\zeta_{\varepsilon}$, where $\zeta_{\varepsilon}(x) = \varepsilon^{-d-1} \zeta(t/\varepsilon, x/\varepsilon)$, nonnegative $\zeta\in C^{\infty}_{0}$ has integral one and $\zeta(t, x) = 0$ for $t\geq0$. Also introduce $I^{\varepsilon}$ by replacing $u$ in the definition of $I$ with $u^{(\varepsilon)}$. Of course, $I^{\varepsilon}\to I$ as $\varepsilon \downarrow 0$ and by Theorem 5.11

Observe that if $\varepsilon$ is small enough and $(s, y)\in C_{R'}$, then $\partial_{t} u^{(\varepsilon)}(s, y) = (I_{C_{R}}\partial_{t}u) *\zeta_{\varepsilon}(s, y)$. Similar formulas are valid for $D^{2}u^{(\varepsilon)}$ and by Minkowski's inequality (the norm of a sum is less then the sum of norms) we have

Since in the last integral $C_{R'}-\varepsilon(s, y)\subset C_{R}$ if $\varepsilon$ is small enough, it follows that for small $\varepsilon$

which yields the desired result.

Acknowledgments

The author brings his gratitude to the referees for their useful comments.

Conflict of interest

The author declares no conflict of interest.