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Besov and Triebel–Lizorkin spaces on metric spaces: Embeddings and pointwise multipliers ☆

Research paper by Wen Yuan

Indexed on: 22 Apr '17Published on: 11 Apr '17Published in: Journal of Mathematical Analysis and Applications



Abstract

In this paper, we obtain the Franke–Jawerth embedding property of Hajłasz–Besov and Hajłasz–Triebel–Lizorkin spaces on a measure metric space (X,d,μ)(X,d,μ) which is Ahlfors regular with dimension “Q  ”. As applications, we show that, when (X,d,μ)(X,d,μ) is doubling and satisfies an Ahlfors lower bound condition with Q  , then the Hajłasz–Besov space <img height="18" border="0" style="vertical-align:bottom" width="58" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si113.gif">Np,qs(X) with p∈(Q,∞]p∈(Q,∞], <img height="22" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si145.gif">s∈(Qp,1] and q∈(0,∞]q∈(0,∞] and the Hajłasz–Triebel–Lizorkin space <img height="18" border="0" style="vertical-align:bottom" width="61" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si100.gif">Mp,qs(X) with p∈(Q,∞)p∈(Q,∞), <img height="22" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si145.gif">s∈(Qp,1] and <img height="23" border="0" style="vertical-align:bottom" width="92" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si11.gif">q∈(QQ+s,∞] are algebras under pointwise multiplication and, moreover, when XX is Ahlfors Q  -regular, we characterize the class of all pointwise multipliers on the Hajłasz–Triebel–Lizorkin space <img height="18" border="0" style="vertical-align:bottom" width="61" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si100.gif">Mp,qs(X) for p∈(Q,∞)p∈(Q,∞), <img height="22" border="0" style="vertical-align:bottom" width="67" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si145.gif">s∈(Qp,1] and <img height="23" border="0" style="vertical-align:bottom" width="92" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022247X17303669-si11.gif">q∈(QQ+s,∞] by its related uniform space.