# Real interpolation with variable exponent

## TL;DR

This paper develops a theory of real interpolation for function spaces with variable exponents, establishing foundational properties and linking it to fixed exponent cases, with applications to variable Besov and Lorentz spaces.

## Contribution

It introduces and analyzes the real interpolation method for variable exponent spaces, extending classical theory and providing new tools for variable Besov and Lorentz spaces.

## Key findings

- Established basic properties of real interpolation with variable exponents.
- Showed reduction to fixed exponent cases under certain conditions.
- Applied the theory to interpolate variable Besov and Lorentz spaces.

## Abstract

We present the real interpolation with variable exponent and we prove the basic properties in analogy to the classical real interpolation. More precisely, we prove that under some additional conditions, this method can be reduced to the case of fixed exponent. An application, we give the real interpolation of variable Besov and Lorentz spaces as introduced recently in Almeida and H\"ast\"o (J. Funct. Anal. 258 (5) 1628--2655, 2010) and L. Ephremidze et al. (Fract. Calc. Appl. Anal. 11 (4) (2008), 407--420).