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Real-variable characterizations of new anisotropic mixed-norm Hardy spaces
A Lusin type result
Department of Mathematics, University of Trento, via Sommarive 14, 38123 Trento, Italy |
$ A\subset {\mathbb{R}}^n $ |
$ k $ |
$ x\in A $ |
$ P_x: {\mathbb{R}}^n\to {\mathbb{R}} $ |
$ k+1 $ |
$ \begin{equation*} {\rm{ap}}\lim\limits_{x\to a}\frac{(D^\alpha P_x)(x)-(D^\alpha P_a)(x)}{\vert x-a\vert} = 0 \end{equation*} $ |
$ \alpha\in {\mathbb{N}}^n $ |
$ \vert\alpha\vert\leq k $ |
$ a\in A $ |
$ \varepsilon >0 $ |
$ \varphi\in C^{k+1}( {\mathbb{R}}^n) $ |
$ {\mathcal L}^n \bigg(A\setminus \bigcap\limits_{\vert\alpha\vert\leq k+1}\{x\in A : D^\alpha\varphi (x) = (D^\alpha P_x)(x)\}\bigg)\leq\varepsilon. $ |
References:
[1] |
B. Bojarski, Differentiation of Measurable Functions and Whitney-Luzin Type Structure Theorems, Helsinki University of Technology Institute of Mathematics Research Reports, Vol. A572, 2009. Available from: http://math.tkk.fi/reports/a572.pdf. |
[2] |
B. Bojarski,
Sobolev spaces and averaging I, Proc. A. Razmadze Math. Inst., 164 (2014), 19-44.
|
[3] |
B. Bojarski, P. Hajlasz and P. Strzelecki,
Improved $C^{k, \lambda}$ approximation of higher order Sobolev functions in norm and capacity, Indiana Univ. Math. J., 51 (2002), 507-540.
doi: 10.1512/iumj.2002.51.2162. |
[4] |
S. Delladio,
A note on some topological properties of sets with finite perimeter, Glasg. Math. J., 58 (2016), 637-647.
doi: 10.1017/S0017089515000385. |
[5] |
S. Delladio,
The tangency of a $C^1$ smooth submanifold with respect to a non-involutive $C^1$ distribution has no superdensity points, Indiana Univ. Math. J., 68 (2019), 393-412.
doi: 10.1512/iumj.2019.68.7549. |
[6] |
L. C. Evans and R. F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of
Functions, Studies in Advanced Math., CRC Press, 1992. |
[7] |
H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. |
[8] |
R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, 1995. |
[9] |
R. Shakarchi and E. M. Stein, Real Analysis (measure theory, integration and Hilbert spaces),
Princeton University Press, Princeton and Oxford, 2005. |
[10] |
W. P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer-Verlag, 1989.
doi: 10.1007/978-1-4612-1015-3. |
show all references
References:
[1] |
B. Bojarski, Differentiation of Measurable Functions and Whitney-Luzin Type Structure Theorems, Helsinki University of Technology Institute of Mathematics Research Reports, Vol. A572, 2009. Available from: http://math.tkk.fi/reports/a572.pdf. |
[2] |
B. Bojarski,
Sobolev spaces and averaging I, Proc. A. Razmadze Math. Inst., 164 (2014), 19-44.
|
[3] |
B. Bojarski, P. Hajlasz and P. Strzelecki,
Improved $C^{k, \lambda}$ approximation of higher order Sobolev functions in norm and capacity, Indiana Univ. Math. J., 51 (2002), 507-540.
doi: 10.1512/iumj.2002.51.2162. |
[4] |
S. Delladio,
A note on some topological properties of sets with finite perimeter, Glasg. Math. J., 58 (2016), 637-647.
doi: 10.1017/S0017089515000385. |
[5] |
S. Delladio,
The tangency of a $C^1$ smooth submanifold with respect to a non-involutive $C^1$ distribution has no superdensity points, Indiana Univ. Math. J., 68 (2019), 393-412.
doi: 10.1512/iumj.2019.68.7549. |
[6] |
L. C. Evans and R. F. Gariepy, Lecture Notes on Measure Theory and Fine Properties of
Functions, Studies in Advanced Math., CRC Press, 1992. |
[7] |
H. Federer, Geometric Measure Theory, Springer-Verlag, 1969. |
[8] |
R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, 1995. |
[9] |
R. Shakarchi and E. M. Stein, Real Analysis (measure theory, integration and Hilbert spaces),
Princeton University Press, Princeton and Oxford, 2005. |
[10] |
W. P. Ziemer, Weakly Differentiable Functions, GTM 120, Springer-Verlag, 1989.
doi: 10.1007/978-1-4612-1015-3. |
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