A NULLSTELLENSATZ FOR AMOEBAS

More documents

Recommendations

Info

542 ASTALA, CLOP, MATEU, OROBITG, and URIARTE-TUERO mapping f : \ E → C admits a K-quasiregular extension to . In this definition, as in the analytic setting, we may replace L ∞ () by BMO() to get a close variant of the problem. The sharpness of the bounds in equation (1.2) determines the index 2/(K + 1) as the critical dimension in both the L ∞ and the BMO quasiregular removability problems. In fact, Iwaniec and Martin [15] previously conjectured that in R n , n ≥ 2, sets with Hausdorff measure H n/(K+1) (E) = 0 are removable for bounded K- quasiregular mappings. A preliminary positive answer for n = 2 was described in [6]. Generalizing this, in the present article, we show that, surprisingly, for K>1, one can do even better. We have the following improved Painlevé removability. THEOREM 1.2 Let K>1, and suppose that E is any compact set with H 2/(K+1) (E) σ -finite. Then E is removable for all bounded K-quasiregular mappings. The theorem fails for K = 1 since, for instance, the line segment E = [0, 1] is not removable for bounded analytic functions. For the converse direction, the article [4] finds for every t>2/(K + 1) non-Kremovable sets with dim(E) = t. We also make an improvement here and construct compact sets with dimension precisely equal to 2/(K + 1) yet not removable for some bounded K-quasiregular mappings (for details, see Theorem 5.1). Theorems 1.1 and 1.2 are closely connected via the classical Stoïlow factorization, which tells (see [6], [18]) that in planar domains, K-quasiregular mappings are precisely the maps f representable in the form f = h ◦ φ, where h is analytic and φ is K-quasiconformal. Indeed, the first step in proving Theorem 1.2 is to show that for a general K-quasiconformal mapping φ, one has H 2/(K+1) (E) σ -finite ⇒ H 1( φ(E) ) σ -finite. However, this conclusion is not enough since there are rectifiable sets of finite length (such as E = [0, 1]) which are nonremovable for bounded analytic functions. Therefore, in addition, we need to establish that such “good” sets of positive analytic capacity actually behave better also under quasiconformal mappings. That is, we show that up to a set of zero length, F 1-rectifiable ⇒ dim ( φ −1 (F ) ) > 2 K + 1 (for details and a precise formulation, see Corollary 3.2).
DISTORTION OF HAUSDORFF MEASURES AND REMOVABILITY 543 The article is structured as follows. In Section 2, we deal with the quasiconformal distortion of Hausdorff measures and of other set functions. In Section 3, we study the quasiconformal distortion of 1-rectifiable sets. Section 4 gives the proof for the improved Painlevé removability theorem for K-quasiregular mappings and other related questions. Finally, in Section 5, we describe a construction of nonremovable sets. 2. Absolute continuity There are several natural ways to normalize the quasiconformal mappings φ : C → C. In this article, we mostly use the principal K-quasiconformal mappings (i.e., mappings that are conformal outside a compact set and are normalized by φ(z) − z = O(1/|z|) as |z| →∞). It is shown in Astala’s article [4] that for all K-quasiconformal mappings φ : C → C, |φ(E)| ≤C |E| 1/K , (2.1) where C is a constant that depends on the normalizations. By scaling, we may always arrange diam ( φ(E) ) = diam(E) ≤ 1, (2.2) and then C = C(K) depends only on K. In order to achieve the result (2.1), one first reduces to the case where the set E is a finite union of disks. Second, applying Stoïlow factorization methods, the mapping φ is written as φ = h ◦ φ 1 , where both h, φ 1 : C → C are K-quasiconformal mappings, so that φ 1 is conformal on E and h is conformal in the complement of the set F = φ 1 (E). Here, one obtains the right conclusion for φ 1 , |φ 1 (E)| ≤C |E| 1/K , by including φ 1 in a holomorphic family of quasiconformal mappings. Further, one shows in [4, page 50] that under the special assumption where h is conformal outside of F ,wehave |h(F )| ≤C |F |, (2.3) where the constant C still depends only on K. In searching for absolute continuity properties of other Hausdorff measures under quasiconformal mappings, such a decomposition seems unavoidable, and this leads one to look for counterparts of (2.3) for Hausdorff measures H t or Hausdorff contents M t . Here, we establish the result for the dimension t = 1.
Page 1 and 2:
A NULLSTELLENSATZ FOR AMOEBAS KEVIN
Page 3 and 4:
A NULLSTELLENSATZ FOR AMOEBAS 409
Page 5 and 6:
A NULLSTELLENSATZ FOR AMOEBAS 411 A
Page 7 and 8:
A NULLSTELLENSATZ FOR AMOEBAS 413 c
Page 9 and 10:
A NULLSTELLENSATZ FOR AMOEBAS 415 I
Page 11 and 12:
Page 13 and 14:
A NULLSTELLENSATZ FOR AMOEBAS 419 C
Page 15 and 16:
A NULLSTELLENSATZ FOR AMOEBAS 421 f
Page 17 and 18:
A NULLSTELLENSATZ FOR AMOEBAS 423 a
Page 19 and 20:
A NULLSTELLENSATZ FOR AMOEBAS 425 N
Page 21 and 22:
Page 23 and 24:
A NULLSTELLENSATZ FOR AMOEBAS 429 4
Page 25 and 26:
A NULLSTELLENSATZ FOR AMOEBAS 431 O
Page 27 and 28:
A NULLSTELLENSATZ FOR AMOEBAS 433 g
Page 29 and 30:
A NULLSTELLENSATZ FOR AMOEBAS 435 z
Page 31 and 32:
A NULLSTELLENSATZ FOR AMOEBAS 437 P
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
A NULLSTELLENSATZ FOR AMOEBAS 443 C
Page 39 and 40:
A NULLSTELLENSATZ FOR AMOEBAS 445 R
Page 41 and 42:
448 DAVID GINZBURG The method we us
Page 43 and 44:
450 DAVID GINZBURG Let ν denote a
Page 45 and 46:
452 DAVID GINZBURG Proof The proof
Page 47 and 48:
454 DAVID GINZBURG correspond to th
Page 49 and 50:
456 DAVID GINZBURG Next, consider t
Page 51 and 52:
458 DAVID GINZBURG To state the fol
Page 53 and 54:
460 DAVID GINZBURG check that up to
Page 55 and 56:
462 DAVID GINZBURG where L τ (¯s)
Page 57 and 58:
464 DAVID GINZBURG To define the li
Page 59 and 60:
466 DAVID GINZBURG cuspidality of t
Page 61 and 62:
468 DAVID GINZBURG immediately foll
Page 63 and 64:
470 DAVID GINZBURG expansion. Here
Page 65 and 66:
472 DAVID GINZBURG Notice that S l
Page 67 and 68:
474 DAVID GINZBURG values of m ′
Page 69 and 70:
476 DAVID GINZBURG For 1 ≤ i ≤
Page 71 and 72:
478 DAVID GINZBURG 2m rows, we have
Page 73 and 74:
480 DAVID GINZBURG left to right. C
Page 75 and 76:
482 DAVID GINZBURG unipotent orbit
Page 77 and 78:
484 DAVID GINZBURG for every choice
Page 79 and 80:
486 DAVID GINZBURG is equal to L(σ
Page 81 and 82:
488 DAVID GINZBURG Here f π (h) is
Page 83 and 84: 490 DAVID GINZBURG character ψ Um
Page 85 and 86: 492 DAVID GINZBURG Proof The proof
Page 87 and 88: 494 DAVID GINZBURG in the above ref
Page 89 and 90: 496 DAVID GINZBURG THEOREM 7 The ir
Page 91 and 92: 498 DAVID GINZBURG In the above, th
Page 93 and 94: 500 DAVID GINZBURG is GL 2m+1 × SO
Page 95 and 96: 502 DAVID GINZBURG [GP] S. GELBART
Page 97 and 98: A CHARACTERIZATION OF SUBSPACES AND
Page 111 and 112: DEGREE GROWTH OF MEROMORPHIC SURFAC
Page 131 and 132: DISTORTION OF HAUSDORFF MEASURES AN
Page 133: DISTORTION OF HAUSDORFF MEASURES AN
Page 165 and 166: 574 MARCHÉ and NARIMANNEJAD them w
Page 167 and 168: 576 MARCHÉ and NARIMANNEJAD 1.1. P
Page 169 and 170: 578 MARCHÉ and NARIMANNEJAD 2.1. T
Page 171 and 172: 580 MARCHÉ and NARIMANNEJAD Figure
Page 173 and 174: 582 MARCHÉ and NARIMANNEJAD Figure
Page 175 and 176: 584 MARCHÉ and NARIMANNEJAD Defini
Page 177 and 178: 586 MARCHÉ and NARIMANNEJAD [A2] [
show all

A NULLSTELLENSATZ FOR AMOEBAS

Create successful ePaper yourself

Delete template?

Save as template?