Abstract
We characterize complex measures <svg style="vertical-align:-2.29482pt;width:9.6374998px;" id="M1" height="9.6750002" version="1.1" viewBox="0 0 9.6374998 9.6750002" width="9.6374998" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,9.675)"> <g transform="translate(72,-64.26)"> <text transform="matrix(1,0,0,-1,-71.95,66.6)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝜇</tspan> </text> </g> </g> </svg> on the unit ball of <svg style="vertical-align:-0.17555pt;width:17.200001px;" id="M2" height="11.625" version="1.1" viewBox="0 0 17.200001 11.625" width="17.200001" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,11.625)"> <g transform="translate(72,-62.7)"> <text transform="matrix(1,0,0,-1,-71.95,62.92)"> <tspan style="font-size: 12.50px; " x="0" y="0">ℂ</tspan> </text> <text transform="matrix(1,0,0,-1,-63.14,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> </g> </g> </svg>, for which the general Toeplitz operator <svg style="vertical-align:-4.74141pt;width:18.012501px;" id="M3" height="17.112499" version="1.1" viewBox="0 0 18.012501 17.112499" width="18.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,17.1125)"> <g transform="translate(72,-58.31)"> <text transform="matrix(1,0,0,-1,-71.95,63.09)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝑇</tspan> </text> <text transform="matrix(1,0,0,-1,-63.07,68.09)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝛼</tspan> <tspan style="font-size: 8.75px; " x="-2" y="8.1300001">𝜇</tspan> </text> </g> </g> </svg> is bounded or compact on the analytic Besov spaces <svg style="vertical-align:-4.74141pt;width:44.012501px;" id="M4" height="16.6" version="1.1" viewBox="0 0 44.012501 16.6" width="44.012501" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,16.6)"> <g transform="translate(72,-58.72)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑝</tspan> </text> <text transform="matrix(1,0,0,-1,-58.47,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝔹</tspan> </text> <text transform="matrix(1,0,0,-1,-45.85,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> <text transform="matrix(1,0,0,-1,-41.01,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg>, also on the minimal Möbius invariant Banach spaces <svg style="vertical-align:-3.20526pt;width:44.125px;" id="M5" height="14.6875" version="1.1" viewBox="0 0 44.125 14.6875" width="44.125" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.6875)"> <g transform="translate(72,-60.25)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝐵</tspan> </text> <text transform="matrix(1,0,0,-1,-63.25,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">1</tspan> </text> <text transform="matrix(1,0,0,-1,-58.37,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">(</tspan> <tspan style="font-size: 12.50px; " x="4.1634989" y="0">𝔹</tspan> </text> <text transform="matrix(1,0,0,-1,-45.76,60.37)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> <text transform="matrix(1,0,0,-1,-40.91,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">)</tspan> </text> </g> </g> </svg> in the unit ball <svg style="vertical-align:-3.20526pt;width:16.75px;" id="M6" height="14.4625" version="1.1" viewBox="0 0 16.75 14.4625" width="16.75" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,14.4625)"> <g transform="translate(72,-60.43)"> <text transform="matrix(1,0,0,-1,-71.95,63.67)"> <tspan style="font-size: 12.50px; " x="0" y="0">𝔹</tspan> </text> <text transform="matrix(1,0,0,-1,-63.5,60.55)"> <tspan style="font-size: 8.75px; " x="0" y="0">𝑛</tspan> </text> </g> </g> </svg>.
Highlights
Let Bn be the unit ball of the n-dimensional complex Euclidean space Cn
We denote the class of all holomorphic functions on the unit ball Bn by H Bn
If μ is a Ap Bn, p -Carleson measure, the Toeplitz operator Tμα is bounded on Bp Bn spaces if and only if Pα μ w is a Bp Bn, p -Carleson measure
Summary
Let Bn be the unit ball of the n-dimensional complex Euclidean space Cn. We denote the class of all holomorphic functions on the unit ball Bn by H Bn. Boundedness and compactness of general Toeplitz operators Tμα on the α-Bloch Bα D spaces have been investigated in 7 on the unit disk D for 0 < α < ∞. In 8 , the authors extend the Toeplitz operator Tμα to Bα Bn in the unit ball of Cn and completely characterize the positive Borel measure μ such that Tμα is bounded or compact on Bα Bn with 1 ≤ α < 2. We will extend the general Toeplitz operators Tμα to Bp Bn in the unit ball o f Cn and completely characterize the positive Borel measure μ such that Tμα is bounded or compact on the Bp Bn spaces with 2n < p < ∞. A positive Borel measure μ on the unit ball Bn is said to be a Carleson measure for the Bergman space Apα Bn if. The letter C will denote a positive constant, possibly different on each occurrence
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