Control of the bilinear indicator cube testing property
We show that the \(α\)-fractional bilinear indicator/cube testing constant \(BICT_T^α ( σ ,ω ) _QP^n _E,FQ1Q_σ Q_ω _FT_σ^α ( 1_E) ω ,\) defined for any \(α\)-fractional singular integral \(T^α \) on \(R^n\) with \(0<α <n\), is controlled by the classical \(α\)-fractional Muckenhoupt constant \(A_2^α ( σ ,ω)\), provided the product measure \(σ × ω\) is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding \(2(n-α )\). Moreover, this control is sharp within the class of diagonally reverse doubling product measures. In fact, every product measure \(μ × μ\), where \(μ\) is an Ahlfors-David regular measure \(μ\) with exponent \(n-α\), has diagonal exponent \(2( n-α )\) and satisfies \(A_2^α ( μ ,μ )<∞\) and \(BICT_I^α ( μ ,μ )=∞\), which has implications for the \(L^2\) trace inequality of the fractional integral \(I^α\) on domains with fractional boundary. When combined with the main results in arXiv:1906.05602, 1907.07571 and 1907.10734, the above control of \(BICT_T^α \) for \(α>0\) yields a \(T1\) theorem for doubling weights with appropriate diagonal reverse doubling, i.e. the norm inequality for \(T^α\) is controlled by cube testing constants and the \(α\)-fractional one-tailed Muckenhoupt constants \(A_2^α \) (without any energy assumptions), and also yields a corresponding cancellation condition theorem for the kernel of \(T^α \), both of which hold for arbitrary \(α\)-fractional Calderón-Zygmund operators \(T^α \). We do not know if the analogous result for \(BICT_H(σ,ω )\) holds for the Hilbert transform \(H\) in case \(α=0\), but we show that \(BICT_H^dy (σ ,ω)\) is not controlled by the Muckenhoupt condition \(A_2^α ( ω ,σ )\) for the dyadic Hilbert transform \(H^dy \) and doubling weights \(σ ,ω \). , number=2, journal=Annales Fennici Mathematici, author=Sawyer, Eric T. and Uriarte-Tuero, Ignacio, year=2021, month=Sep., pages=1105–1122