It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $ n $-dimensional rectangle and the inductive application of the Sobolev trace embedding results. The new method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions with the upgrade of pointwise regularity to continuity. Remarkably, it also presents the largest class of weakly differentiable functions where the generalized Newton-Leibniz formula holds.
Citation: Ugur G. Abdulla. Generalized Newton-Leibniz formula and the embedding of the Sobolev functions with dominating mixed smoothness into Hölder spaces[J]. AIMS Mathematics, 2023, 8(9): 20700-20717. doi: 10.3934/math.20231055
It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. The proof is based on the generalization of the Newton-Leibniz formula to the $ n $-dimensional rectangle and the inductive application of the Sobolev trace embedding results. The new method is applied to prove the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions with the upgrade of pointwise regularity to continuity. Remarkably, it also presents the largest class of weakly differentiable functions where the generalized Newton-Leibniz formula holds.
| [1] | S. L. Sobolev, On a theorem of functional analysis, Mat. Sbornik, 4 (1938), 471–497. |
| [2] | R. A. Adams, J. F. Fournier, Sobolev spaces, Elsevier, 2003. |
| [3] | C. B. Morrey, Jr, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43, 1938,126–166. https://doi.org/10.2307/1989904 |
| [4] | S. M. Nikol'skii, On boundary properties of differentiable functions of several variables, Russ. Acad. Sci., 146 (1962), 542–545. |
| [5] | S. M. Nikol'sky, On stable boundary values of differentiable functions of several variables, Mat. Sb., 61 (1963), 224–252. |
| [6] | O. V. Besov, V. P. Il'in, S. M. Nikol'skii, Integral representations of functions and imbedding theorems, Washington: Winston Sons, 1978. |
| [7] | H. J. Schmeisser, H. Triebel, Topics in Fourier analysis and function spaces, Wiley, 1987. |
| [8] | J. Vybiral, Function spaces with dominating mixed smoothness, Diss. Math., 436, 2006, 1–73. |
| [9] | H. Triebel, Function spaces with dominating mixed smoothness, EMS Series of Lectures in Mathematics, Zürich: Eur. Math. Soc., 2019. |
| [10] | F. J. Hickernell, I. H. Sloan, G. W. Wasilkowski, On tractability of weighted integration over bounded and unbounded regions in $\mathbb{R}^s$, Math. Comput., 73 (2004), 1885–1901. |
| [11] | T. Tao, Sobolev Spaces, 2009, Available from: https://terrytao.wordpress.com/2009/04/30/245c-notes-4-sobolev-spaces/ |
| [12] | V. K. Nguyen, W. Sickel, Isotropic and dominating mixed besov spaces-a comparison, 2016, https://arXiv.org/abs/1601.04000 |
| [13] | H. Triebel, Theory of Function Spaces II, Birkhäuser Basel, 1992. https://doi.org/10.1007/978-3-0346-0419-2 |
| [14] |
V. K. Nguyen, W. Sickel, Pointwise multipliers for Sobolev and Besov spaces of dominating mixed smoothness, J. Math. Anal. Appl., 452 (2017), 62–90. https://doi.org/10.1016/j.jmaa.2017.02.046 doi: 10.1016/j.jmaa.2017.02.046
|
| [15] | R. A. Adams, Reduced Sobolev inequalities, Can. Math. Bull., 31 (1988), 159–167. https://doi.org/10.4153/CMB-1988-024-1 |
Ugur G. Abdulla. Generalized Newton-Leibniz formula and the embedding of the Sobolev functions with dominating mixed smoothness into Hölder spaces[J]. AIMS Mathematics, 2023, 8(9): 20700-20717. doi: 10.3934/math.20231055
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