On the boundedness of the classical fractional integral operator in the variable exponent Hardy-amalgam spaces

Abstract

In this paper, we study the mapping properties of the fractional integral operator \(I_{\gamma }\) \( (0<\gamma <d)\) on the Hardy-amalgam spaces with variable exponents. More precisely, we show that if the measurable exponent functions \(p\left( \cdot \right) \) and q satisfy log-Hölder continuity conditions locally and at infinity, then the fractional integral operator of order \(\gamma \), \(I_{\gamma }\) maps \(\mathcal {H}^{p\left( \cdot \right) ,q}\) spaces continuously into \(\mathcal {H}^{p_{1}\left( \cdot \right) ,q_{1}}\) spaces on \(\mathbb {R}^{d}\) under the assumptions \(\frac{1}{ p_{1}\left( x\right) }=\frac{1}{p\left( \cdot \right) }-\frac{\gamma }{d}\) and \(\frac{1}{q_{1}}=\frac{1}{q}-\frac{\gamma }{d}\).