Anisotropic versions of the Brezis-Van Schaftingen-Yung approach at $s=1$ and $s=0$

In 2014, Ludwig showed the limiting behavior of the anisotropic Gagliardo s-seminorm of a function f as s → 1 and s → 0, which extend the results due to Bourgain-Brezis-Mironescu(BBM) and Maz’ya-Shaposhnikova(MS) respectively. Recently, Brezis, Van Schaftingen and Yung provided a different approach by replacing the strong L norm in the Gagliardo s-seminorm by the weak L quasinorm. They characterized the case for s = 1 that complements the BBM formula. The corresponding MS formula for s = 0 was later established by Yung and the first author. In this paper, we follow the approach of Brezis-Van Schaftingen-Yung and show the anisotropic versions of s = 1 and s = 0. Our result generalizes the work by Brezis, Van Schaftingen, Yung and the first author and complements the work by Ludwig.

[1]  L. Grafakos.  Classical Fourier Analysis , 2010 .

[2]  Petru Mironescu,et al.  Limiting embedding theorems forWs,p whens ↑ 1 and applications , 2002 .

[3]  Deping Ye,et al.  Anisotropic Sobolev Capacity with Fractional Order , 2014, Canadian Journal of Mathematics.

[4]  M. Ludwig.  Anisotropic fractional Sobolev norms , 2013, 1304.0703.

[5]  Po-Lam Yung,et al.  Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail , 2021 .

[6]  G. Paouris,et al.  Relative entropy of cone measures and Lp centroid bodies , 2009, 0909.4361.

[7]  C. Bianchini,et al.  Some overdetermined problems related to the anisotropic capacity , 2018, Journal of Mathematical Analysis and Applications.

[8]  A. Figalli,et al.  Symmetry results for critical anisotropic p-Laplacian equations in convex cones , 2019, Geometric and Functional Analysis.

[9]  M. Ludwig.  Anisotropic fractional perimeters , 2013, 1304.0699.

[10]  R. E. Castillo,et al.  An Introductory Course in Lebesgue Spaces , 2016 .

[11]  J. Bourgain,et al.  Another look at Sobolev spaces , 2001 .

[12]  R. Ambartzumian.  Stochastic and integral geometry , 1987 .

[13]  E. Lutwak,et al.  Blaschke-Santaló inequalities , 1997 .

[14]  M. Ludwig.  Ellipsoids and matrix-valued valuations , 2003 .

[15]  A. Figalli,et al.  Sharp stability theorems for the anisotropic Sobolev and log-Sobolev inequalities on functions of bounded variation , 2013 .

[16]  Po-Lam Yung,et al.  A new formula for the $L^p$ norm , 2021, 2102.09657.

[17]  Minkowski Valuations,et al.  Minkowski Valuations , 2004 .

[18]  Arkady Poliakovsky.  Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung , 2021, Journal of Functional Analysis.

[19]  E. Lutwak,et al.  A new ellipsoid associated with convex bodies , 2000 .

[20]  Vladimir Maz'ya,et al.  On the Bourgain, Brezis, and Mironescu Theorem Concerning Limiting Embeddings of Fractional Sobolev Spaces , 2003 .

[21]  Bo'az Klartag,et al.  Centroid Bodies and the Logarithmic Laplace Transform - A Unified Approach , 2011, 1103.2985.

[22]  C. Bianchini,et al.  Wulff shape characterizations in overdetermined anisotropic elliptic problems , 2017, 1703.07111.

[23]  Vladyslav Yaskin,et al.  Centroid bodies and comparison of volumes , 2006 .

[24]  E. Valdinoci,et al.  Hitchhiker's guide to the fractional Sobolev spaces , 2011, 1104.4345.

[25]  Erwin Lutwak,et al.  Orlicz centroid bodies , 2010 .

[26]  M. Squassina,et al.  On anisotropic Sobolev spaces , 2017, Communications in Contemporary Mathematics.

[27]  G. Paouris.  Concentration of mass on convex bodies , 2006 .

[28]  Po-Lam Yung,et al.  A surprising formula for Sobolev norms , 2021, Proceedings of the National Academy of Sciences.

[29]  Grigoris Paouris,et al.  A stability result for mean width of Lp-centroid bodies , 2007 .

[30]  Ali Maalaoui,et al.  Characterizations of anisotropic high order Sobolev spaces , 2018, Asymptot. Anal..

[31]  E. Lutwak,et al.  The Cramer-Rao inequality for star bodies , 2002 .

[32]  G. Leng,et al.  On the Lp affine isoperimetric inequalities , 2011 .

[33]  Erwin Lutwak,et al.  Moment-entropy inequalities , 2004 .

[34]  Pierre-Louis Lions,et al.  Convex symmetrization and applications , 1997 .

[35]  Franz E Schuster,et al.  General $L_p$ affine isoperimetric inequalities , 2008, 0809.1983.

[36]  Lukas Parapatits,et al.  SL(n)‐covariant Lp‐Minkowski valuations , 2012, J. Lond. Math. Soc..

[37]  D. Ma.  Asymmetric anisotropic fractional Sobolev norms , 2014, 1410.5940.

[38]  Po-Lam Yung,et al.  A new formula for the L norm , 2021 .

[39]  C. Bianchini,et al.  An overdetermined problem for the anisotropic capacity , 2015, Calculus of Variations and Partial Differential Equations.

[40]  C. Villani,et al.  A MASS-TRANSPORTATION APPROACH TO SHARP SOBOLEV AND GAGLIARDO-NIRENBERG INEQUALITIES , 2004 .