On Weak and Strong Sharp Weighted Estimates of Square Function

We reduce here end-point estimates for one singular operator (namely for dyadic square function) to Monge–Ampère equations with drift. The spaces are weighted spaces, and therefore the domain, where we solve our PDE is non-convex. If we are in the end-point situation our goal is either to find a logarithmic blow-up of the norm estimate from below, or to prove that there is upper estimate of the norm without logarithmic blow-up. For two other singular operators: the martingale transform and the Hilbert transform the logarithmic blowup was demonstrated in [5]. This disproved the so-called weak Muckenhoupt conjecture, while [6] and [7] disproved the strong Muckenhoupt conjecture for these operators. The difference with the current situation is that for the martingale transform and for the Hilbert transform (for their weak type estimates in weighted spaces) the end-point exponent is 1, interesting weights were A1 weights, and the abovementioned logarithmic blow-up concerns log[w]A1 . For the weak type estimate of the square function the end-point exponent is 2, and the abovementioned logarithmic blow-up concerns log[w]A2 . We show that for weak type test condition there is no logarithmic blow-up, which is strikingly different from the upper estimate in [3] and [1], where logarithm of [w]A2 is present. 1. Notations and results We reduce here end-point estimates for one singular operator (namely for dyadic square function) to Monge–Ampère equations with drift. The spaces are weighted spaces, and therefore the domain, where we solve our PDE is non-convex. If we are in the end-point situation our goal is either to find a logarithmic blow-up of the norm estimate from below, or to prove that there is upper estimate of the norm without logarithmic blow-up. For two other singular operators: the martingale transform and the Hilbert transform the logarithmic blow-up was demonstrated in [5]. This disproved the so-called weak Muckenhoupt conjecture, while [6] and [7] disproved the strong Muckenhoupt conjecture for these operators. (Disproving a weak conjecture is a stronger result than disproving the strong conjecture.) The difference with the current situation is that for the martingale transform and for the Hilbert transform (for their weak type estimates in weighted spaces) the end-point exponent is 1, interesting weights were A1 weights, and the mentioned logarithmic blow-up concerns log[w]A1 . For the weak type estimate of the square function the end-point exponent is 2, and the above mentioned logarithmic blow-up concerns log[w]A2 . We show that for weak type test condition there is no logarithmic blow-up, which is strikingly different from the upper estimate in [3] and [1], where logarithm of [w]A2 is present. In the paper [3] the weak weighted norm of the square function from L2(w) to L2,∞(w) was estimated from above by √ [w]A2 log[w]A2 , this was improved to √ [w]A2 √ log[w]A2 . 2010 Mathematics Subject Classification. 42B20, 42B35, 47A30. FN is partially supported by the NSF grant DMS . AV is partially supported by the NSF grant DMS-1265549 and by the Hausdorff Institute for Mathematics, Bonn, Germany. 1 2 P. IVANISVILI, F. NAZAROV, AND A. VOLBERG It is easy to conjecture that this estimate is sharp. However, below we show that if one tests the norm on characteristic functions of cubes (intervals) then logarithmic term disappears. In particular, if there were a T1 theorem for weak type theorems, then we would conclude that logarithmic term totally disappears, and one would be able to improve the estimates √ [w]A2 √ log[w]A2 of [3], [1] to a simple √ [w]A2 (which cannot be improved by an obvious examples of one-singular-point weights). It is interesting that for the square function estimate the end-point exponent is 2 and one should work with A2 weights. The square function operator is much less singular than the martingale transform and the Hilbert transform. This is why the game happens around √ [w]A2 . We recall the reader that for the martingale transform and the Hilbert transform the play happens around [w]A1 and the end-point exponent is 1. Let us recall that for those operators the estimate from below logarithmically blows up. It has been shown in [5] that ‖L1(w) → L1,∞(w)‖-norm of both martingale and Hilbert transform can have the estimate from below of the order [w]A1(log[wA1 ]) ε, for ε > 0. In fact, it has been shown that any ε ∈ (0, 0.2) would work (and ε ∈ (0, 2/7) would work for another type of martingale transform). In what follows, given w ∈ A2, we consider for constants: 1) ‖L2(w)→ L2,∞(w)‖-norm of the square function operator on test functions, 2) the full ‖L2(w) → L2,∞(w)‖-norm of the square function operator, 3) ‖L2(w)→ L2(w)‖-norm of the square function operator on test functions, 4) the full ‖L2(w) → L2(w)‖-norm of the square function operator. The strong norms in 3) and 4) are known of course, and we include them into consideration by two reasons. The first reason is that by Chebyshev inequality one can estimate the weak norms by the strong ones, and the second reason is to show how useless they are for the sharp weak estimate via Chebyshev inequality, they are of order [w]A2 , while the weak estimates are small perturbations of √ [w]A2 . One disclaimer: the reader should notice that everywhere below we are working not with norm (strong or weak) but with squares of the norms. 1.1. Four constants. We consider the square function transform. To do that we consider the dyadic (for simplicity) lattice D and call its elements by I, J, . . . . We consider martingale differences (the symbol ch(J) denotes the dyadic children of J)