L. Kokilashvili

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

ON FOURIER MULTIPLIERS AND THE SOLVABILITY OF CERTAIN SUPERLINEAR PDE

The main goal of this report is to present two-weighted estimates for the multipliers of Fourier transforms and their application to the multidimensional Ricatti equation.

Let S(R^{n}) be Schwartz space of rapidly decreasing functions. Denote by F (φ) the Fourier transform of φ ∈ S(R^{n}). Let the operator K defined by the Fourier transform equation

F (Kφ) = mF (φ), φ ∈ S.

The following statement is true:

Theorem. Let the function v_{j}, j = 1, 2, satisfy the conditions

(i) v_{j} ∈ L^{1}(R^{1}), j = 1, 2.

(ii) ∫_{R^{1}} \frac{1}{v_{2}(x)|x|^{(1−α)2}} dx < ∞.

Moreover, assume that

m(λ) = ∫_{−∞}^{λ} dμ,

where μ is a finite measure on R^{1}. Then the following inequality

∫_{R^{1}} |Kf(x)|^{2} \frac{1}{w(x)} dx ≤

≤ ( ∫_{R^{1}} \frac{1}{v_{2}(x)|x|^{(1−α)2}} dx)( ∫_{R^{1}} |f(x)|^{2} \frac{1}{v_{1}(x)} dx)

holds, where w =^{def} v_{1} ∗ v_{2}.

We intend the application to the solvability problem for equation

−Δu = |∇u|^{p} + f, 1 < p < ∞,

to discuss.