Boundedness of integral operators of double phase. (English) Zbl 1531.31011
Summary: Our aim in this note is to establish a Sobolev-type inequality and Trudinger-type inequality for fractional maximal and Riesz potential operators in the framework of general double phase functionals given by
\[
\varphi (x,t) = \varphi_1 (t) + \varphi_2 (b(x)t), \, x\in \mathbb{R}^n, \, t\geqslant 0,
\]
where \(\varphi_1\), \(\varphi_2\) are positive convex functions on \((0, \infty)\) and \(b\) is a nonnegative function on \([0,\infty)\) which Hölder continuous of order \(\theta \in (0,1]\).
MSC:
| 31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Riesz potentials; Sobolev inequality; Trudinger inequality; Orlicz spaces; double phase functionalsReferences:
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| [29] | September 22, 2022) Yoshihiro Mizuta Graduate School of Advanced Science and Engineering Hiroshima University Higashi-Hiroshima 739-8521, Japan e-mail: yomizuta@hiroshima-u.ac.jp |
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