Herz Spaces Meet Morrey Type Spaces and Complementary Morrey Type Spaces
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(2020) 26:74
Herz Spaces Meet Morrey Type Spaces and Complementary Morrey Type Spaces Humberto Rafeiro1
· Stefan Samko2
Received: 28 September 2019 / Revised: 29 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce local and global generalized Herz spaces. As one of the main results we show that Morrey type spaces and complementary Morrey type spaces are included into the scale of these Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized Herz spaces with application to Morrey type spaces and their complementary spaces, based on the mentioned inclusion. Keywords Function spaces · Herz spaces · Morrey type spaces · Sublinear operators · Maximal function · Singular operators Mathematics Subject Classification Primary 46E30; Secondary 47B38
1 Introduction The well-known Herz spaces introduced in [15] attracted a lot of attention last decade as an example of decomposition spaces widely used in applications ([7]), see for instance studies of Herz spaces in [8,10,14,18,21,23], including results in variable exponent setting in [2,16,20,26–30,33], see also references therein.
Communicated by Winfried Sickel.
B
Humberto Rafeiro [email protected] Stefan Samko [email protected]
1
Department of Mathematical Sciences, College of Science, United Arab Emirates University, P.O. Box 15551 Al Ain, Abu Dhabi, United Arab Emirates
2
Department of Mathematics, University of Algarve, Campus de Gambelas, Faro, Portugal 0123456789().: V,-vol
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Let α ∈ R, 1 ≤ p < ∞, 1 ≤ q < ∞. Homogeneous Herz spaces are defined by the norm
f K˙ p,q (Rn ) := α
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩k∈Z
⎛
⎜ 2kαq ⎝
2k−1 0, max(2a,τ ) 0, if |x − y| > 2|y − z|, |K (x, y) − K (x, z)| ≤ C |x − y|n+σ |x − ξ |σ |K (x, y) − K (ξ, y)| ≤ C , σ > 0, if |x − y| > 2|x − ξ |. |x − y|n+σ
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Example 4.5 Let
β t α0 ln et 0 , t < 1 ω(t) = α t ∞ (ln(et))β∞ , t ≥ 1. Then any sublinear operator satisfying the size condition (15) and bounded in L p (Rn ) p,q p,q is bounded in the generalized Herz spaces H ω (Rn ) and H ω,0 (Rn ) under the conditions: −
n n n n < α0 < , − < α∞ < . p p p p
Corollary 2 Let 1 < p < ∞, 1 < q < ∞, and ω ∈ M(R+ ). Then the maximal operator M, the Hardy operators H and H, and Calderón-Zygmund singular operator with standard kernel T are bounded in: p,q
(a) M ω (Rn ) if − np < m 0 (ω) ≤ M0 (ω) < 0 and − np < m ∞ (ω) ≤ M∞ (ω) < 0, (b)
M p,q (Rn ) ω,0
if 0 < m 0 (ω) ≤ M0 (ω)
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