Orlicz spaces bounds for special classes of hyperbolic sums
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- 作者：Dimitrios Karslidis dimkars@math.ubc.ca
- 关键词：Small ball inequality ; Littlewood&ndash ; Paley inequalities ; Haar functions ; Hyperbolic sums ; Orlicz spaces
- 刊名：Journal of Mathematical Analysis and Applications
- 出版年：2017
- 出版时间：15 May 2017
- 年：2017
- 卷：449
- 期：2
- 页码：1302-1320
- 全文大小：438 K
- 卷排序：449
文摘
Let R=R1×⋯×RdR=R1×⋯×Rd denote a dyadic rectangle in the unit cube [0,1]d[0,1]d, d≥3d≥3. Let hRhR be the L∞L∞-normalized Haar function supported on R. In [10], the conjectured signed small ball inequality,‖∑|R|=2−nαRhR‖∞≳nd2,whereαR∈{±1}, was proven under the additional assumption that the coefficients also satisfy the splitting property, αR=αR1⋅αR2×⋯×RdαR=αR1⋅αR2×⋯×Rd with αR1,αR2×⋯×Rd∈{±1}αR1,αR2×⋯×Rd∈{±1}. We give another proof of this result, using a duality argument. Based on this approach, we also show‖∑|R|=2−nαRhR‖exp(La)≳nd2−1a,2≤a<∞ for any integer n≥1n≥1 and any choice of coefficients {αR}⊂{−1,1}{αR}⊂{−1,1} which satisfy the splitting property. The above inequality has been conjectured for general coefficients αR∈{−1,1}αR∈{−1,1} in d≥3d≥3. These bounds are investigated further for more general coefficients {αR}⊂{−1,1}{αR}⊂{−1,1}. The proof of the sharpness of the L∞L∞-lower bound of hyperbolic sums with coefficients satisfying the “splitting property” is also provided.