Herz-type Besov spaces of variable smoothness and integrability

Abstract

The aim of this paper is to study properties of Besov-type spaces with variable smoothness. We show that these spaces are characterized by the ϕ-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces.

Mixed-Norm Herz Spaces and Their Applications in Related Hardy Spaces

“…We refer the reader to [66,40,41,26,62] for various Herz-Hardy spaces and to [27,77] for the Herz-Hardy spaces of variable smoothness and integrability. For more progress on Herz-type spaces, we refer the reader to [86,107,108,109,25,20,22] for Herz-type Besov spaces and to [74,106,110,111,19,21] for Herz-type Triebel-Lizorkin spaces.…”

Estimates of bilinear pseudodifferential operators associated to bilinear Hörmander classes in Besov and Triebel–Lizorkin spaces with variable exponents

“…Since variable exponent function spaces have widely used in many fields such as electrorheological fluid [ 43 ], differential equations [ 19 , 23 , 41 ] and image restoration [ 9 , 22 , 29 , 34 , 46 ], many classical constant exponent function spaces have been generalized to variable exponent setting, such as variable exponent Bessel potential spaces [ 4 , 21 ], variable Hajłasz–Sobolev spaces [ 5 ], variable exponent Besov and Triebel–Lizorkin spaces [ 3 , 13 , 16 , 27 , 30 , 31 , 49 ], variable exponent Hardy spaces [ 38 , 56 ], variable exponent Morrey spaces [ 2 ], variable exponent Herz spaces [ 1 , 26 , 44 ], variable exponent Herz-type Hardy spaces [ 18 , 28 , 48 ], variable exponent Herz–Morrey Hardy spaces [ 50 ], variable exponent Herz-type Besov and Triebel–Lizorkin spaces [ 14 , 17 , 45 , 52 ], variable exponent Morrey-type Besov and Triebel–Lizorkin spaces [ 20 ], Herz–Morrey-type Besov and Triebel–Lizorkin spaces with variable exponents [ 15 ], Triebel–Lizorkin-type spaces with variable exponents [ 57 ], variable weak Hardy spaces [ 53 ], Besov-type spaces with variable smoothness and integrability [ 58 ], variable integral and smooth exponent Triebel–Lizorkin spaces associated with a non-negative self-adjoint operator [ 51 ], variable exponent Hardy spaces associated with operators [ 55 ], and variable Hardy spaces associated with operators [ 54 , 59 , 60 ]. For the boundedness of integral operators in variable function spaces, we recommend [ 32 ] and [ 33 ].…”

Besov-type spaces of variable smoothness on rough domains

“…Besov spaces of variable smoothness (and various extensions thereof) have been intensively studied in the last 30 years. Among the fundamental works we mention [1], [21], [18], [17], [19], [20], [10], [11], [12], [13], [14], [3], [4], [33], [34], [22], [15], and [29]. We also note the recent paper [35], which introduces a new Besov-type space, which has not only variables smoothness, but also in a sense a 'variable structure'.…”