Strichartz estimates and admissible exponents

For free Schrödinger on R to the d, call a pair (q,r) admissible when both exponents are at least two, they satisfy two over q plus d over r equals d over two, and the triple (q,r,d) is not (2,infinity,2). The homogeneous estimate bounds the L to the q time, L to the r space norm of U(t)f over the full real-time line by a constant times the L² norm of f. The pair (infinity,2) is the energy estimate and belongs to this family.

The exceptional two-dimensional pair satisfies scaling but fails the general estimate, showing that dimensional balance alone is insufficient. In three dimensions the endpoint (2,6) does hold; it must not be discarded merely because q equals two. In one dimension, (4,infinity) is admissible and valid. Different dispersive equations have different admissibility relations, and compact spatial domains need their own statements, sometimes with losses or time restrictions. Every application should name its equation, spatial domain, and endpoint convention.

For two admissible pairs, the retarded Duhamel operator maps an appropriate dual source space into the solution spacetime space. A sufficient version used here takes both time exponents strictly greater than two, allowing the energy time exponent infinity, and excludes the exceptional pair already stated. If the source pair is (q-tilde,r-tilde), its Hölder conjugates q-tilde-prime and r-tilde-prime describe the forcing norm. The time integral runs from the initial time to t, not over all times without order.

The proof combines the dispersive bound, unitarity, duality, and a time-ordering argument. Endpoint forcing estimates require their own justified theorem; they are not obtained by simply substituting q=2 into every nonendpoint proof. To use the estimate nonlinearly, the source |u|²u must land in the required dual norm and its difference must obey a corresponding Lipschitz bound. The linear theorem is a tool for that nonlinear accounting, not a statement that every power nonlinearity is automatically controlled.

For one-dimensional cubic NLS, work in the intersection of continuous L²-valued functions and L⁸-time/L⁴-space functions on [0,T]. The displayed source bound, with its difference version, makes the Duhamel correction at most a constant times square root of T times the cube of a ball radius. The Lipschitz coefficient has the same time factor times radius squared. Choosing a short interval gives a local L² construction, rougher than the H¹ algebra construction used earlier on the torus.

At the one-dimensional mass-critical quintic power, the admissible pair (6,6) gives the source L to the six fifths norm directly from five factors in L⁶, with no extra interval-length power. A critical local proof instead exploits smallness of the free evolution's L⁶ spacetime norm on a sufficiently short interval for the particular datum, together with a carefully chosen contraction domain. One cannot manufacture the missing subcritical time power by relabeling exponents. This contrast makes scaling's warning concrete in an actual nonlinear estimate.