Integrability for Hardy operators of double phase

Yoshihiro Mizuta; Tetsu Shimomura

doi:10.4171/zaa/1732

JournalszaaVol. 42, No. 3/4pp. 375–402

Integrability for Hardy operators of double phase

Yoshihiro Mizuta
Hiroshima University, Japan
Tetsu Shimomura
Hiroshima University, Japan

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Abstract

We establish Hardy–Sobolev and Hardy–Trudinger inequalities in weighted Orlicz spaces on $R^{n}$ . As an application, we prove Hardy–Sobolev and Hardy–Trudinger inequalities in the framework of general double phase functionals given by

φ_{p} (x, t) = φ_{1} (t^{p}) + φ_{2} ((b (x) t)^{p}), x \in R^{n}, t \geq 0,

where $p > 1$ , $φ_{1}, φ_{2}$ are positive convex functions on $(0, \infty)$ and $b$ is a non-negative function on $[0, \infty)$ which is Hölder continuous of order $θ \in (0, 1]$ .

Cite this article

Yoshihiro Mizuta, Tetsu Shimomura, Integrability for Hardy operators of double phase. Z. Anal. Anwend. 42 (2023), no. 3/4, pp. 375–402

DOI 10.4171/ZAA/1732