Orlicz spaces bounds for special classes of hyperbolic sums

Let R=R1×⋯×Rd denote a dyadic rectangle in the unit cube [0,1]d, d≥3. Let hR be the 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×⋯×Rd with 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≥1 and any choice of coefficients {αR}⊂{−1,1} which satisfy the splitting property. The above inequality has been conjectured for general coefficients αR∈{−1,1} in d≥3. These bounds are investigated further for more general coefficients {αR}⊂{−1,1}. The proof of the sharpness of the L∞-lower bound of hyperbolic sums with coefficients satisfying the “splitting property” is also provided.