Research proof index

Hölder's inequality, binary case.

Hölder's inequality, binary case.

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. In the project’s cancellative metric-space setting, assume the associated nontangential operators satisfy the required uniform $L^2$ bound. If $F$ and $G$ are measurable and $f$ is measurable with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, then the Carleson operator obeys the restricted estimate

$\int_G^+ \mathcal{C}_K f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

A modular form tends to its value at infinity as z → i∞.

For $0\le t\le1$, the reciprocal of $1-2^{-t}$ is controlled in the extended nonnegative reals by

$(1-2^{-t})^{-1}\le 2t^{-1}.$

If g(t₀) = 0 and deriv g t₀ < 0, then g is negative shortly after t₀.